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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bundle theorem ： ウィキペディア英語版 Bundle theorem In geometry, the bundle theorem is in the simplest case a statement on six circles and eight points in the real Euclidean plane. In general it is a property of a Möbius plane that is fulfilled by ''ovoidal'' Möbius planes only. The bundle theorem should not be confused with Miquel's theorem. An ovoidal Möbius plane in real Euclidean space may be considered as the geometry of the plane sections of an egglike surface, like a sphere or an ellipsoid or a half of a sphere clued to a suitable half of an ellipsoid or the surface with equation $x^4+y^4+z^4=1$, .... If the egglike surface ist just a sphere one gets the space model of the classical real Möbius plane, the ''circle geometry'' on the sphere. The essential property of an ovoidal Möbius plane is the existence of a space model via an ovoid. An ovoid in a 3-dimensional projective space is a set of points, which a) is intersected by lines in 0, 1, or 2 points and b) its tangents at an arbitrary point covers a plane (tangent plane). The geometry of an ovoid in projective 3-space is a Möbius plane, called ovoidal Möbius plane. The point set of the geometry consists of the points of the ovoid and the curves (cycles) are the plane sections of the ovoid. A suitable stereographical projection shows: For any ovoidal Möbius plane there exists a plane model.〔Hartmann, p. 63.〕 In the classical case the plane model is the ''geometry of the circles and lines'' (any line is completed by a point $\backslash infty$). The bundle theorem has a planar and a spacial interpretation. In the planar model there may be lines involved. The proof of the bundle theorem is performed within the spacial model. For any ovoidal Möbius plane $\backslash mathfrak\; M$ the bundle theorem holds: Bundle theorem: *If for different points $A\_1,A\_2,A\_3,A\_4,\; B\_1,B\_2,B\_3,B\_4$ five of the six quadruples The proof is a consequence of the following considerations, which use essentially the fact, that three planes in a 3-dimensional projective space intersect in a single point: #The planes containing the cycles $c\_,c\_,c\_$ intersect in a point $P$. Hence $P$ is the intersection point of the lines (in space !) $A\_2B\_2,\backslash \; A\_4B\_4$. #The planes containing the cycles $c\_,c\_,c\_$ intersect in a point $P\text{'}$. Hence $P\text{'}$ is the intersection point of the lines $A\_2B\_2,\backslash \; A\_4B\_4$, too. This yields: a) $P=P\text{'}$ and b) $A\_1B\_1,\backslash \; A\_3B\_3$ intersect at point $P$, too. The last statement means: $A\_1,B\_1,A\_3,B\_3$ are concyclic. The planes involved have point $P$ in common, they are elements of a ''bundle'' of planes. The importance of the bundle theorem was shown by Jeff Kahn. Theorem of Kahn: A Möbius plane is ovoidal if and only if it fulfills the bundle theorem ,.〔Kahn, p. 62.〕 The bundle theorem has an analogous meaning for Möbius planes as the Theorem of Desargues for projective planes. From the bundle theorem follows the existence of a) a skewfield (division ring) and b) an ovoid. If the more strict theorem of Miquel holds, the skewfield is even commutative (field) and the ovoid is a quadric. Remark: There are Möbius planes, which are not ovoidal.〔Hartmann, p. 64.〕 Remark: For ovoidal Laguerre planes there exists a bundle theorem with an analogous meaning, too.〔Hartmann, p. 78.〕 == References == 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bundle theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.